Integrable Functions with Equal Integrals on Sub-Sigma-Algebra are A.E. Equal

Theorem

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $\GG$ be a sub-$\sigma$-algebra of $\Sigma$.

Let $f, g: X \to \overline \R$ be $\GG$-integrable functions.

Suppose that, for all $G \in \GG$:

$\displaystyle \int_G f \rd \mu = \int_G g \rd \mu$

Then $f = g$ $\mu$-almost everywhere.